Advanced Math / Equivalent Expressions

Equivalent Exponential Expressions

Equivalent exponential expressions use exponent rules, such as adding exponents when multiplying powers with the same base and multiplying exponents when raising a power to a power. Rewriting expressions with a common base is often the key step.

Least you need to know

$a^m\cdot a^n=a^{m+n}$ for the same base $a$.
$\frac{a^m}{a^n}=a^{m-n}$ for $a\ne0$.
$(a^m)^n=a^{mn}$.
A fractional exponent such as $a^{1/2}$ represents a square root.

Key notation

$2^3\cdot2^5$ — A product of powers with the same base
$(x^4)^3$ — A power raised to another power

Worked example

To simplify $\frac{7^{10}}{7^4}$, subtract the exponents.
$\frac{7^{10}}{7^4}=7^{10-4}=7^6$.

Common mistakes

Only add exponents when multiplying powers with the same base.
When a power is raised to another power, multiply the exponents.
Do not treat $a^m+a^m$ as $a^{2m}$; combine the coefficient instead.

How to recognize it

The problem asks for an equivalent expression.
The expression contains repeated bases.
The expression contains a quotient or a power raised to a power.

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